AP Calculus Derivative and Integral Formulas
(d/dx) tan x
sec²x
∫cos x dx
sin x + C
∫ sec² x dx
tan x + C
(d/dx) f(u)
u' f'(u)
(d/dx) ln u
u'/u
(d/dx) (uv)
uv' + vu'
(d/dx) sec x
sec x tan x
∫xⁿ dx
( [xⁿ⁺¹]/[n+1] ) + C, n≠-1
∫ [ (u')/ (a²+u²) ] du
(1/a) arctan (u/a) + C
(d/dx) t/b
(bt' - tb')/(b²)
∫ tan x dx
- ln |cosx| +C
∫sin x dx
-cos x + C
∫ csc² x dx
-cot x + C
∫csc x cot x dx
-csc x + C
(d/dx) cotx
-csc²x
∫cot x dx
-ln |csc x| + C
(d/dx) cos x
-sin x
∫ sec x tan x dx
sec x + C
(d/dx) log base a of x
1/ x ln a
(d/dx) arctan x
1/(1+x²)
(d/dx) ln x
1/x
(d/dx) arcsin x
1/√(1-x²)
A = (1/2) w (h1 + h2)
Area of Trapezoid
(slope between two points) [f(b)-f(a)]/[b-a]
Average Rate of Change
f ' (c)
Instant Rate of Change
a function f that is continuous on [a,b] takes on every y-value between f(a) and f(b)
Intermediate Value Theorem
f ' (c) = f(b) - f(a) / (b-a)
Mean Value Theorem
∫ f ' (x) dx = f(b) - f(a)
Net Change/ Fundamental Theorem of Calculus
d/dx ∫ f(t) dt = f(v)v' - f(u)u'
Second Fundamental Theorem of Calculus
f(b) = f(a) + ∫ f ' (x) dx
Start Plus Net Change/Accumulation
Volume of a Washer
V = π ∫ (R²-r²) dx
Volume of a disc
V = π∫r² dx
Volume of the Cross Section
V = ∫ A dx
∫a^x dx
[(a^x)/ln a] + C
(d/dx) a^x
a^x ln a
∫[ (u')/ (√a²-u²) ] du
arcsin (u/a) + C
(d/dx) sin x
cos x
(d/dx) e^x
e^x
∫ e^x dx
e^x + C
∫tan x dx
ln |secx| + C
∫ cot x dx
ln |sinx| + C
∫ (u'/u) dx
ln |u| + C
∫ 1/x dx
ln |x| + C
(d/dx) xⁿ
nxⁿ⁻¹
speed =
|v(t)|
Total distance
∫ |v(t)|dt
